Using the entropy monotonicity of Proposition REF  the uniform \(\kappa \) -noncollapsing estimate follows from Perelman's original argument ([1]}, cf. [2]} Theorem 6.29). Now assume we have a solution to the normalized flow on \([0,\infty )\)  which satisfies a uniform curvature estimate. By the uniform \(\kappa \) -noncollapsing it follows that the volumes of unit balls are uniformly bounded below along the flow, and there is a uniform lower bound on the injectivity radius. Since the volume form is bounded above pointwise by \(F_t^m\) , the total volume is bounded above by \(\int _M F_t^m = (2 \pi c_1(M, J))^m\) . It follows that there is a uniform upper bound for the diameter along the flow. With these estimates in place, it follows from Cheeger-Gromov compactness theory (cf. [3]} Theorem 2.3 for our simplified case) that for any sequence \(\lbrace t_j\rbrace  \rightarrow \infty \)  there exists a subsequence, still denoted \(\lbrace t_j\rbrace \) , such that \((M, g_{t_j})\)  converges in Cheeger-Gromov sense to a smooth limiting Riemannian manifold \((M, g_{\infty })\) , where the limiting manifold is still \(M\)  due to the diameter upper bound. Also, it follows from ([4]}, [2]} Ch. 5) that \(H\)  and all its covariant derivatives are uniformly bounded along the flow, and thus by choosing a further subsequence \(H\)  can be assumed to converge to \(H_{\infty }\) . We claim that by choosing a further subsequence the generalized Kähler structure also converges, which requires establishing \(C^{\infty }\)  estimates for \(I\)  and \(J\) . First, the \(g\) -norms of \(I\)  and \(J\)  are uniformly bounded since \(g\)  is compatible with both. By a standard argument using the fact that the Bismut \(\nabla ^{B,I}\)  preserves \(I\) , and that the coefficients of this connection are already converging by the discussion above, it follows that there are uniform \(C^{\infty }\)  estimates for \(I\) , and the same argument using \(\nabla ^{B,J}\)  yields estimates for \(J\) . It follows that we can choose a further subsequence such that \((M, g_{t_j}, I_{t_j}, J)\)  converges in Cheeger-Gromov sense to a limiting generalized Kähler manifold \((M, g_{\infty }, I_{\infty }, J_{\infty })\) .
Now note that by rescaling the result of Proposition REF  holds for the normalized flow, setting \(\tau \equiv 1\) . This also implies that the corresponding functional \(\mu \)  is monotone increasing. Since the solution is nonsingular, there are uniform bounds on \(\mu \)  and all of its derivatives. It thus follows that for an arbitrary sequences of times \(\lbrace t_j\rbrace  \rightarrow \infty \)  one has
\(\lim _{j \rightarrow \infty } \left| \frac{d}{dt} \mu (g_{t_j}, H_{t_j}) \right| = 0.\) 

For such a sequence \(\lbrace t_j\rbrace \) , let \((M, g_{\infty }, I_{\infty }, J_{\infty })\)  denote the generalized Kähler limit constructed as above. It follows from the entropy monotonicity formula that the limiting torsion \(H_{\infty }\)  must vanish, and furthermore \(g_{\infty }\)  is a shrinking soliton. Since \(H_{\infty }\)  vanishes, the pairs \((g_{\infty }, I_{\infty })\)  and \((g_{\infty }, J_{\infty })\)  are both Kähler; this shows that the symplectic 2-form \(F_{\infty }\)  taming \(I_{\infty }\)  and \(J_{\infty }\)  is parallel with respect to \(g_{\infty }\) , and hence must be a \((1, 1)\) -form by the Fano condition of \((M, J_{\infty })\) . It then follows that we must have \(I_{\infty } = J_{\infty }\) .
